Time Value of Money

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MGMT 235

Dr. Sharp @2 4

Review Fundamentals of Valuation
These class notes review this material and also provide some help for a financial calculator. It also has some self-test questions and problems. Class notes are necessarily brief. See any principles of finance book for a more extensive explanation.
Eugene F. Brigham, Joel F. Houston Fundamentals of financial management HG 4 2! B!!"3 #""$ %oss, Stephen &, 'ester(iel), an) Jor)an Fundamentals of corporate finance HG 4 2! .%!** #""5

PART I: Sin le Sum.
Time !alue of "oney: #now this terminolo y and notation F+ 3+ i t Future +alue 3resent +alue %ate per perio) 5 o( time perio)s ,#-i.t Future +alue /nterest Fa0tor 1F+/F2 #4,#-i.t 3resent +alue /nterest Fa0tor 13+/F2

$uestion: %hy are &'(i) and &'(i)t called interest factors* +nswer: '. Start with simple arithmetic problem on interest: Ho6 mu0h 6ill 7# , pla0e) in a 8an9 a00ount pa:ing 5; per :ear 8e 6orth 0ompoun)e) annuall:< +nswer: 3rin0ipal - /nterest 7# , - 7# , = . 5 > 7# ,5 ,. -actor out the .'/0///. # , = ,#. 5. > 7# ,5 1. This leaves &'./2) as the factor. #. Fin) the ?alue o( 7# , earning 5; interest per :ear a(ter two :ears. Start 6ith the amount a(ter one :ear an) multipl: 8: the (a0tor (or ea0h :ear. 3&mount a(ter one :ear4 = ,#. 5. > 37# , = ,#. 5.4 = ,#. 5. > 7# , = ,#. 5.2 > 7##, 25. So &'(i)t 5 .

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+. -uture !alue Fin) the ?alue o( 7# , in # :ears. The in?estment earns 5; per :ear.

F+ > 7# , 6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i. F+ > 7# , 6,#. 5.6,#. 5.6,#. 5.6,#. 5.6,#. 5.6,#. 5.6,#. 5.6,#. 5.6,#. 5.6,#. 5. F+ > 7# , = ,#. 5.# > 7# , = #.!2$$" > 7#!,2$" Fin) the ?alue o( 7# , in # :ears. The in?estment earns $; (or (our :ears an) then earns 4; (or the remaining si= :ears. F+ > 7# , 6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i. F+ > 7# , 6,#. $.6,#. $.6,#. $.6,#. $.6,#. 4.6,#. 4.6,#. 4.6,#. 4.6,#. 4.6,#. 4. F+ > 7# , = ,#. $.4 = ,#. 4.! F+ > 7#*,2#4.53

B.

7resent !alueC

Same i)ea, 8ut 8egin at the en). %earrange the Future ?alue eDuation to loo9 li9e thisC
3+ > F+E 1,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.6,#-i.2 3+ > F+ E ,#-i.t 122

8xample: 9ow much do I need to invest at :; per year0 in order to have .'/0/// in<<. a. =ne year: 7! 5'/0/// > &'./:) 5 .?0,2?.,@ b. Two years: 7! 5 .'/0/// > &'./:) > &'./:) =A .'/0/// > &'./:), 5 .:02B1 c. Ten years 7! 5 .'/0/// > &'./:)'/ 5 .'/0/// > ,.'2:? 5 .C0@1, C. Aate of Aeturn ST+AT %IT9 S+"8 A8D+TI=E9SI7: F+ > 3+ = ,#-i.t Sol?e (or i.

,#-i.t >F+43+. #-i > ,F+43+.#4t #4t i > ,F+43+. F#.
$uestion: &n in?estor )eposits 7# , . Ten :ears later it is 6orth 7#*,"# . 'hat rate o( return )i) the in?estor earn on the in?estment< Solution: 7#*,"# > 7# , = ,#-i.# ,#-i.# > 7#*,"# 4# , > #.*"# ,#-i. > ,#.*"# . #4# > #. ! i > . ! > !. ;

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F. -indin the -uture !alue Fin) the ?alue o( 7# , to)a: at the en) o( # perio)s at 5; per perio). '. Scientific Calculator: Gse 1:=2 : > ,#-i. > #. 5 an) = >t> # . #. Enter #. 5. 2. 3ress 3yx4. 1. Enter the e=ponent. 4. Enter 1>2. 5. Multipl: result 8: 7# , .

#8G A8D+TI =E9SI7 :

,. Spreadsheet:

1. -inancial calculator. Gou may need to input somethin like this.
Spe0i(i0 (un0tions ?ar:. Be sure to 0onsult the 0al0ulatorHs manualIIIIII
n 1B2 # i 1/4J%2 5 3+ # , 3MT F+ <

E=T8: The future value will be ne ative0 indicate an opposite direction of cash flow. #. Set the 0al0ulator (reDuen0: to on0e per perio). 2. Enter negati?e num8ers using the 1-4F2 9e:, not the su8tra0tion 9e:. 3. Be sure the 0al0ulator is set in the EBD mo)e. 8. -undamental Idea.

$uestion: 'hat is the ?alue o( an: (inan0ial asset< +nswer: The present ?alue o( its e=pe0te) 0ash (lo6s.

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-. -indin the 7resent !alue
Fin) the present ?alue o( 7# , to 8e re0ei?e) at the en) o( # perio)s at $; per perio).

a. S0ienti(i0 Aal0ulator #8G A8D+T I=E9S
Scientific Calculator: Hse 3yx 4 where y 5 './: and x 5 -'0-,0 or -'/.

#. Enter #. $. 2. 3ress 3yx4 1. Enter the e=ponent as a negati?e num8er 4. Enter 1>2. 5. Multipl: result 8: 7# , .

8. Sprea)sheet

c. -inancial calculator. Gou may need to input somethin like this.
Spe0i(i0 (un0tions ?ar:. Be sure to 0onsult the 0al0ulatorsH manualIIIIII
n 1B2 0. # i 1/4J%2 $ 3+ # , 3MT F+ <

The present value will be ne ative0 to indicate the opposite direction of cash flow.

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I. -indin the 3 eometric avera e4 rate of return:

#8G A8D+TI =E9SI7:
4. 3ress 354. 5. Su8tra0t #

Scientific Calculator To find i0 use 3yx 4 and 3'Jx4.
#. Enter '.B?'/, 2. 3ress 3yx4 1. Enter the e=ponent '/ then press 3'Jx4

,. Spreadsheet

1. -inancial Calculator. &Gour financial calculator may differ. Consult your manual.)
n 1B2 # i 1/4J%2 < 3+ F# , 3MT F+ #*,"#



&ns6er i > !; $uestion: Today your stock is worth .2/0///. Gou invested .20/// in the stock ': years a o. %hat avera e annual rate of return 3i4 did you earn on your investment*
+nswer: '1.@C@;.



$uestion: The total percenta e return was C20///>2///5?//;. %hy doesnKt the avera e rate of return equal 2/;0 since ?//;>': 5 2/;*

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9. -HTHA8 !+DH8 %98E A+T8S =- IET8A8ST C9+EI8. -! 5 7! x &'(i') x 8xample: Jou in?est 7# , . During the (irst :ear the in?estment earne) 2 ; (or the :ear. During the se0on) :ear, :ou earne) onl: 4; (or that :ear. Ho6 mu0h is :our original )eposit 6orth at the en) o( the t6o :ears< -! 5 7! x &'(i') x &'(i,) 5 .'/0/// x &'.,/L) x &'./C) 5 .',0C:/. $uestion: The arithmetri0 a?erage rate o( return is #2;, 6hat is the geometri0 a?erage rate o( return< +nswer: &n a?erage rate o( return is a geometri0 a?erage sin0e it is a rate o( gro6th. The #2; is the arithmeti0 a?erage. The geometri0 a?erage rate o( return on the in?estment 6as ##.*;.
i K%

> ,F+43+.#4tF# > ,#2,4$ 4#

.#42F# > .##*#
not gro6 8:

i = ,#.2 . ⋅ ,#. 4. −# = .##*#

Important: &lthough 2 ; an) 4; a?erage to #2;, the 7# , #2;. 17# , = ,#.#2.2> #2,544 E=T 7#2,4$ 2. I.

C="7=HEFIEI 78AI=FS Gp to this point, 6e ha?e use) :ears as the onl: time perio). &0tuall:, all the pre?ious e=amples 0oul) ha?e 8een Duarters, months, or )a:s. The interest rate and time period must correspond.

8xample: 7roblem '. Fin) the ?alue o( 7# , 7roblem ,. Fin) the ?alue o( 7# ,

earning 5; interest per :ear a(ter t6o :ears. earning 5; interest per Duarter a(ter t6o Duarters.

Both pro8lems ha?e same ans6er 7# , = ,#. 5.2 > 7##, 25. 9owever: /n the (irst pro8lem t re(ers to :ears an) i re(ers to interest rate per :ear. /n the se0on) pro8lem t re(er to Duarters an) i to interest rate per Duarter. -!t 5 7! x &'(i)t. t > num8er o( perio)s i > interest (or the perio).

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+lternatively, -!t•m 5 7! x &'(iJm)t•m. m> perio)s per :ear, t> num8er o( :ears, i > the interest per :ear 1&3%2. 8xample: 'hat 6ill 7#, 8e 6orth at the en) o( one :ear 6hen the annual interest rate is #2; 1This is the +7A.2 6hen interest is 0ompoun)e)C +nnuallyC t5' i 5',; F+# > 3+ = ,#-i.# > 7#, = ,#.#2.# > 7#,#2 . 4 4 $uarterly: t5C i 5 1; F+4 > 3+ = ,#-i. > 7#, = ,#. 3. > 7#,#25.5#. "onthly: t5', i 5'; F+#2 > 7#, = ,#. #.#2 > 7#, = ,#.#2!$25. > 7#,#2!.$25. Faily: t51@2 i 5 &',;>1@2) 5 /./1,:BB; F+3!5 > 7#, = ,#. 32$**.3!5> 7#,
i 1/4J%2 #2 3 # . 32$** 3+ #, #, #, #,

= ,#.#2*4*. > 7#,#2*.4*.
3MT F+ < < < <

n 1B2 # 4 #2 3!5

9ow about compoundin at every instant* 8. C=ETIEH=HS C="7=HEFIEI: 3Hsed in Mlack Scholes option pricin model.4 tLm lim # - MMiMM 5 ei t ∞ m m 8xample: 'hat is 7#, 6orth in one :ear i( 0ompoun)e) at #2; 0ontinuousl:.

F+ > 7#, = e.#2 > 7#, = #.#2*4"* > 7#,#2*.5 This is 7. 3 more than )ail: 0ompoun)ing. Try this on your calculator. -ind the ex button. e.', 5 '.',BC? 7resent !alue Interest -actor 5 3e Fi t2 7roblem: 'hat is the present ?alue o( 7# , to 8e re0ei?e) 3 :ears (rom to)a: 0ompoun)e) 0ontinuousl: at # ;<3+ > 7# , = e F.# = 3 > 7# , = .*4 $2>7*,4 $ x Try this on your calculator. -ind the e button. e-/.1 5 /.BC/:,

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7ractice $uiN $uestions: 7! and -! of a Sin le sum.
Aeview 7roblems #. Ho6 mu0h must :ou )eposit to)a: in a 8an9 a00ount pa:ing interest 0ompoun)e) Duarterl:C a. i( :ou 6ish to ha?e 7# , 8. i( :ou 6ish to ha?e 75 , 0. i( :ou 6ish to ha?e 7!, at the en) o( 3 months, i( the 8an9 pa:s 5. ; &3%< &ns6erC 7",$** at the en) o( 24months, i( the 8an9 pa:s $. ;&3%< &ns6erC 742,!*5 at the en) o( #2 months, i( the 8an9 pa:s ". ; &3%< &ns6erC 75,4$"

2. a. 'hat rate o( interest 1&3%2 is the 8an9 0harging :ou i( :ou 8orro6 7**,!5 an) must repa: 7$ , at the en) o( 2 Duarters, i( interest is 0ompoun)e) Duarterl:<
&ns6erC !. ; &3%

8. 'hat rate o( interest 1&3%2 is the 8an9 0harging :ou i( :ou 8orro6 74", 75 , at the en) o( 3 months, i( interest is 0ompoun)e) monthl:<

an) must repa:
&ns6erC $. ; &3%

3. Ho6 mu0h must :ou )eposit to)a: in a 8an9 a00ount pa:ing interest 0ompoun)e) monthl:C at the en) o( # months, i( the 8an9 pa:s 5. ; &3% < &ns6erC 7","5" 8. i( :ou 6ish to ha?eC N!, at the en) o( ! months, i( the 8an9 pa:s ". ; &3% < &ns6erC N5,*3* 0. i( :ou 6ish to ha?eC 7#2, at the en) o( #2 months, i( the 8an9 pa:s !. ; &3% < &ns6erC 7##,3 3 4. /( interest is 0ompoun)e) Duarterl:, ho6 mu0h 6ill :ou ha?e in a 8an9 a00ountC a. i( :ou )eposit to)a: N$, at the en) o( 3 months, i( the 8an9 pa:s 5. ; &3% < &ns6erC N$,# 8. i( :ou )eposit to)a: 7# , at the en) o( ! months, i( the 8an9 pa:s ". ; &3% < &ns6erC 7# ,455 0. i( :ou )eposit to)a: O$ , at the en) o( #2 months, i( the 8an9 pa:s $. ; &3% < &ns6erC O$!,5"5 ). i( :ou )eposit to)a: 75, at the en) o( 24 months, i( the 8an9 pa:s 5. ; &3% < &ns6erC 75,522 a. i( :ou 6ish to ha?eC 7# ,

5. /( interest is 0ompoun)e) monthl:, ho6 mu0h 6ill :ou ha?e in a 8an9 a00ount, a. i( :ou )eposit to)a: N$, 8. i( :ou )eposit to)a: 7# , 0. i( :ou )eposit to)a: O$ , ). i( :ou )eposit to)a: N5, at the en) o( 3 months, i( the 8an9 pa:s 5. ; &3% < &ns6erC N$,# at the en) o( ! months, i( the 8an9 pa:s ". ; &3% < &ns6erC 7# ,45" at the en) o( #2 months, i( the 8an9 pa:s $. ; &3% < &ns6erC O$!,!4 at the en) o( 24 months, i( the 8an9 pa:s 5. ; &3% < &ns6erC N5,525

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!. Jou 8orro6e) 7#,5$4 an) must repa: 72, in e=a0tl: 4 :ears (rom to)a:. /nterest is 0ompoun)e) annuall:. a. 'hat is the interest rate 1&3%2 o( the loan< &ns6er !. ; 8. 'hat e((e0ti?e annual rate 1E&%2 are :ou pa:ing< &ns6er !. ; *. Jou no6 ha?e 7$, in a 8an9 a00ount in 6hi0h :ou ma)e one single )eposit 7$, monthl: o( 7#4$."* e=a0tl: 4 :ears ago. /nterest is 0ompoun)e) monthl:. a. 'hat rate o( interest 1&3%2 is the 8an9 pa:ing< &ns6er # . ; 8. 'hat e((e0ti?e annual rate 1E&%2 is the 8an9 pa:ing< &ns6er # .4*; 7ossibly Eew 7roblems. $. Suppose :ou ma9e an in?estment o( 7#, . This (irst :ear the in?estment returns #2;, the se0on) :ear it returns !;, an) the thir) :ear in returns $;. Ho6 mu0h 6oul) this in?estment 8e 6orth, assuming no 6ith)ra6als are ma)e< &ns6erC
# P,#.#2. = ,#. !. = ,#. $. > 7#,2$2

". 'h: is ,#-i. 0alle) an interest (a0tor< Fa0toring the e=pression 7# ,
# . Suppose :ou ma9e an in?estment o( 7#,

-# ,

=i ># ,

= ,#-i.

Thus ,#-i. is an interest (a0tor.

. This (irst :ear the in?estment returns 5;, the se0on) :ear it returns i. 'rite an e=pression, using i, that represents the (uture ?alue o( the in?estment at the en) o( t6o :ears.
&ns6erC

F+>#,

= ,#. 5. = ,#-i.

##. &n in?estment is 6orth 75 , to)a:. This (irst :ear the in?estment returns ";, the se0on) :ear it returns i. 'rite an e=pression using i that represents the original ?alue o( the in?estment. &ns6erC 3+>5 , E1,#. ". = ,#-i.2 #2. Suppose :ou ma9e an in?estment o( 7&. This (irst :ear the in?estment returns # ;, the se0on) :ear it returns #!;, an) the thir) :ear in returns 2;. Ho6 mu0h 6oul) this in?estment 8e 6orth, assuming no 6ith)ra6als are ma)e< &ns6erC
&P,#.# . = ,#.#!. = ,#. 2.

##. Suppose :ou ma9e an in?estment o( 7# , . This (irst :ear the in?estment returns #5;, the se0on) :ear it returns 2;, an) the thir) :ear in returns # ;. Ho6 mu0h 6oul) this in?estment 8e 6orth at the en) o( three :ears, assuming no 6ith)ra6als are ma)e<

7#2," 3

#2. %e(er to the a8o?e pro8lem. 'hat is the geometri0 a?erage rate o( return<

$.";

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Review Fundamentals of Valuation Part II Multiple 3erio)sC Gne?en an) E?en ,&nnuities. • 3erio)i0 Gne?en Aash Flo6s
'hat is the ?alue o( the (ollo6ing set o( 0ash (lo6s to)a:< The interest rate is $; (or all 0ash (lo6s.
Jear an) Aash Flo6 #C 7 3 2C 7 5 3C 7* 4C 7#



SolutionC Fin) Ea0h 3resent +alue an) &))
3 5 * # + + + = #. $# #. $ 2 #. $ 3 #. $ 4
277.78 428.67 555.68 735.03 = 1997.16

• 3erio)i0 Aash Flo6C E?en 3a:ments
&n annuit: is a le?el series o( pa:ments. For e=ample, (our annual pa:ments, 6ith the (irst pa:ment o00urring e=a0tl: one perio) in the (uture is an e=ample o( an or)inar: annuit:.

7#,
#

7#,
2

7#,
3

7#,
4

+. 7resent value of an annuity:
The present ?alue o( ea0h o( the 0ash (lo6s is the ?alue o( the annuit:. This 0oul) 8e )one one at a time, 8ut this might 8e te)ious. +nnuity 7resent !alue Interest -actor

3+/F& > 1#4,#-i. - #4,#-i.2 - ... - #4,#-i.t2

PVIFA =

∑# 4(1 + i )
j =#

t

j

# − # 4(1 + i ) PVIFA = . i
t

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8xample: %hat is the present value of a C-year annuity0 if the annual interest is 2;0 and the annual payment is .'0///* i 5 2;O 7"T 5 .'0///O t 5CO 7! 5 *
3+ > #, 4,#. 5. - #, 4,#. 5.2 - #, 4,#. 5.3- #, 4,#. 5.4
Don way.

-actor out the sin le sum interest rate factors: 3+ > #, = 1#4,#. 5. - #4,#. 5.2-#4,#. 5.3- #4,#. 5.42 > 3+ > #, = 13+/F& ,4,5;.2 >

Short

Calculate: 3+/F&,4,5;. > #F#4,#-i.t > #F 3+/F4,5; #F .$22* > 3.545"5. i 5; . 5 3+ > #, = 13.54! 2 > 73,54!.

-indin the -uture !alue of an annuity on a:
'. Scientific Calculator. To calculate 7!I-+ usin scientific calculator: -IAST -IEF: T98E -IEF: 3+/F4,5; > #4,#-i.t > #4,#. 5.4 > .$22*

3+/F&,4,5;. > #F#4,#-i.t > #F 3+/F #F .$22* > 3.545"5. i i . 5 5 #, = 13.54! 2 > 73,54!. ,. Hsin a spreadsheet.

1. Hsin a financial calculator0 the 7resent !alue of an annuity.
n 1B2 4 i 1/4J%2 5 3+ < 3MT F# F+

3+> 73,54!.
Eote: "ost financial calculators require i 3IJGA4 to be a percenta e. That is enter a 20 not ./2. 9owever0 8xcel requires ./2 or 2;.

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M. -uture value of an annuity:
• +nnuity -uture !alue Interest -actor F+/F& > 1#- ,#-i. - ,#-i.2 - ... - ,#-i.tF#2.
BoteC Qast 3a:ment earns no interest in

FVIFA = FVIFA =

∑(1 + i )
j=0 t

t −#

t

(1 + i ) − # i

.

8xample: %hat is the future value of a C-year annuity0 if the annual interest is 2;0 and the annual payment is .'0///* i 5 2;O 7"T 5 .'0///O t 5CO -! 5 * 7#, 7#, 7#, = 1#- ,#. 5. - ,#. 5.2 - ,#. 5.32 > = 1F+/F& ,4,5;.2 > = 14.3# #2 > 74,3# .#

-indin -!I-+ '. Hsin scientific calculator: -IAST -IEF: T98E: F+/F > ,#-i.t > ,#. 5.4 > #.2#55 ,#-i.t F# > F+/FF # i i F+/F4,5;F # #.2#55F# > 4.3# # 5; . 5 . Hse

F+/F&,t,i. > F+/F&,t,i. >

,. Hsin a Spreadsheet

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#3

1. Hsin a financial calculator0 the -uture !alue of an annuity:
n 1B2 4 i 1/4J%2 5 3+ 3MT F# F+ <

-! 5 .C01'/ $uestion: Ho6 mu0h 6oul) :ou nee) to )eposit e?er: month in an a00ount pa:ing !; a :ear to a00umulate 8: 7#, , 8: age !5 8eginning at age 2 < DataC F+ > 7#, , i > !;E#2 > .5; per month n > ,!5F2 . = #2 > 45 = #2 > 54 months. 3MT > < &ns6erC 3MT > 73!2.$5

C. A+T8 =- A8THAE =- +E +EEHITG
Jou 8orro6 7! , an) repa: in $ eDual annual installments o( 7#2,"35 6ith the (irst pa:ment ma)e e=a0tl: # :ear later. To the nearest per0ent, 6hat rate o( interest are :ou pa:ing on :our loan< Fifficult without financial calculator. Can use table to find answer to the nearest percent.
υ

Fata: i 5 * 7! 5 .@/0///

7"T5.',0?12

t 5 : years

Aelationship: 7! 5 7"T x 7!I-+&t0 i) '. Solution: &Trial and 8rror with Table) 7!I-+&t0 i) 5 7!J7"T5 .@/0///J',0?12 5 C.@1:@
Table ..... 'C; .... : : .. C.@1:? .. :

Therefore: 7!I-+ 5 C.@1:@. SoP i 5 'C; ,. &Trial and 8rror usin a spreadsheet pro ram)

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#4

1. &Trial and 8rror usin a financial calculator)
n 1B2 $ i 1/4J%2 < ! 3+ 3MT F#2"35 F+

i 5 'C; F. 8xample of +nnuity with quarterly compoundin : &n in?estment o( 73 per Duarter (or ! :ears at annual interest rate o( $;, 0ompoun)e) Duarterl:, 6ill a00umulate 8: the en) o( :ear ! toC SolutionC F+ > < 3MT > 73, t > 24 i > 2; F+ > 3MT = F+/F& ,t, i.. F+ > 73, = 13 .4222 > 7"#,2!!.
n 1B2 24 i 1/4J%2 2 3+ 3MT F3 F+ < .?'0,@@

Aeview 7roblems with solutions. #. This oneC is a t:pi0al mortgage pro8lem. Jou 8orro6 7$ , to 8e repai) in eDual monthl: installments (or 3 :ears. The &3% is ";. 'hat is the monthl: pa:ment< 3+ > 7$ , i > .*5;, t>3! 3MT > < 7$ , > 3MT = #24.2$2 3MT > 7!43.*
F+

n 1B2 3!

i 1/4J%2 .*5

3+ F$

3MT <

2. Tr: this one. Jou ma9e eDual 74 monthl: pa:ments on a loan. The interest rate eDuals #5; &3%, 0ompoun)e) monthl:. The loan is (or #2 :ears. 'hat is the amount o( the loan< &ns6erC 3+ > 72!,!5#
3. %etire 6ith a millionC Ho6 mu0h 6oul) must :ou )eposit monthl: in an a00ount pa:ing !; a :ear 1&3%2, 0ompoun)e) monthl:, to a00umulate 7#, , 8: age !5 8eginning at age 3 < &ns6erC 3MT > 7* #."
n 1B2 42 i 1/4J%2 .5 3+ 3MT < # F+

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#5

C.

Hsin a financial calculator for annuity calculations: Aal0ulate the (uture ?alue o( 7! . per :ear at *; per :ear (or eight :ears.

FV = 60 ∑(1 + .07 )
j=0

7

j

,#. *.$ − # FV = ! ⋅ . * FV = 7!#5.5
n 1B2 $ i 1/4J%2 * 3+ 3MT F! F+ <

F+ > 7!#5.5 5. Aal0ulate the (uture ?alue o( 75 .
n1B2 24 i 1/4J%2 .5

per month at !; &3% (or 24 months
3+ 3MT F5 F+ < -! 5 .'0,'B.@/

!. Aal0ulate the present ?alue o( 75 0ompoun)ing..
n1B2 5 i 1/4J%2 !

per :ear at !; per :ear (or 5 :ears ,monthl:

3+ < 3+>72,# !

3MT F5

F+

B. Gou borrow .20/// and repay the loan with ', equal monthly payments of .2//* Calculate the interest rate per month and the +7A.

n1B2 #2

i 1/4J%2 < i > 2."2; per month. &3% > i = #2 &3% > 2."2; = #2 > 35. 4

3+ 5,

3MT F5

F+

:. 7roblem on inflation. Jou 6ill re0ei?e 7# , )ollars 6hen :ou retire, (ort: :ears (rom to)a:. /( in(lation a?erages 3; per :ear (or the ne=t (ort: :ears, ho6 mu0h 6oul) that amount 8e 6orth measure) in to)a:Rs )ollars< ,Bote, this is not a time ?alue o( mone: pro8lem, 8ut it sol?e) 6ith a similar 0al0ulation. Su0h a)Sustments are ne0essar: to o?er0ome Tmone: illusionU2
υ

Alass Botes

#!

υ υ Solution:

7#

,

E ,#. 3.4 >#

,

E 3.2!2 4 > 7 3 ,!55

F. +nnuity Fue 7#, 7#,
#

7#,
2

7#,
3 4

$uestion: Aompare the pa:ments o( the annuit: )ue, a8o?e, 6ith those o( the or)inar: annuit: earlier. 'hat is the )i((eren0e< Ho6 )oes this )i((eren0e a((e0t its ?alue< +nswer: Ea0h pa:ment in an annuit: )ue o00urs one perio) earlier than it 6oul) in or)inar: annuit:. Both present ?alue an) (uture ?alue o( ea0h pa:ment in an annuit: )ue i( ,#-i. times greater than it 6oul) 8e (or an or)inar: annuit:. $uestion: 'hat is the present ?alue o( the a8o?e (ourF:ear annuit: )ue< 7#, = 1# - #4,#-i. - #4,#-i.2 - #4,#-i.32 5 7#, = ,#-i. = 1#4,#-i. - #4,#-i.2 - #4,#-i.3-#4,#-i.42 5 7#, = ,#-i. = 3+/F& i,4 7! interest factor of an annuity due is: &'(i)67!I-+ -! interest factor of an annuity due is: &'(i)6-!I-+
7roblem. %hat is the present value of an annuity due of five .:// annual payments discounted at '/;* :// x &'.'/)x7!I!+'/;02 5 :// x&'.'/)x 1.B?/B? x 5 :// x C.'@?:B 5 .10112.?

Eote: -inancial calculators have a BEGIN and EN mode. The above assumes the EN mode. If the calculator is set in the BEGIN mode0 it calculates an annuity due. 7roblem. %hat is the present value of an annuity of five annual .:// payments discounted at '/;* The first payment is due in one-half year from today. :// x &'.'/)'J, x7!I!+'/;02 5 :// x&'./C::')x 1.B?/B? x 5 :// x 1.?B2:' 5 10':/.B

Alass Botes

#*

Try the followin practice questions: Aeview 7roblems
#. Suppose :ou are tr:ing to (in) the present ?alue o( t6o )i((erent 0ash (lo6s. Kne is 7# t6o perio)s (rom no6, the other a 7# (lo6 three perio)s (rom no6. 'hi0h o( the (ollo6ing is4are true a8out the )is0ount (a0tors use) to ?alue the 0ash (lo6s< a. The (a0tor (or the (lo6 three perio)s a6a: is al6a:s less than the (a0tor (or the (lo6 that is re0ei?e) t6o perio)s (rom no6. 8. The (a0tor (or the (lo6 three perio)s a6a: is al6a:s more than the (a0tor (or the (lo6 that is re0ei?e) t6o perio)s (rom no6. 0. 'hether one (a0tor is larger than the other 6ill )epen) on the interest rate. ). Sin0e the pa:ments are (or the same amount, the (a0tors 6ill :iel) present ?alues that are the same. e. Bone o( the a8o?e statements are true.

2.

'hat is the present ?alue o( a stream o( 72,5 semiannual pa:ments re0ei?e) at the en) o( ea0h perio) (or the ne=t # :ears< The &3% is !;. a. 3*,#"4 8. 3$,3# 0. 35,$# ). 3!,$$5
pa:ments re0ei?e) at the en) o( ea0h :ear (or the ne=t #

3. 'hat is the (uture ?alue in # :ears o( 7#,5 :ears< &ssume an interest rate o( $;. a. 725,2! 8. 723,4* 0. 72#,*3 ). 7#$,3"5 e. 7#5,

4. Jou are gi?en the option o( re0ei?ing 7#, no6 or an annuit: o( 7$5 per month (or #2 months. 'hi0h o( the (ollo6ing is 0orre0t< a. Jou 0annot 0hoose 8et6een the t6o 6ithout 0omputing present ?alues. 8. Jou 0annot 0hoose 8et6een the t6o 6ithout 0omputing (uture ?alues. 0. Jou 6ill al6a:s 0hoose the lump sum pa:ment. ). Jou 6ill al6a:s 0hoose the annuit:. e. The 0hoi0e :ou 6oul) ma9e 6hen 0omparing the (uture ?alue o( ea0h 6oul) 8e the same as the 0hoi0e :ou 6oul) ma9e 6hen 0omparing present ?alues. 5. Jou open a sa?ings a00ount that pa:s 4.5; annuall:. Ho6 mu0h must :ou )eposit ea0h :ear in or)er to ha?e 75 , (i?e :ears (rom no6< a. 7$,32# 8. 7",!2" 0. 7$,!3! ). 7",#4 e. 7!,5!"

!. Jou are 0onsi)ering an in?estment in a !F:ear annuit:. &t the en) o( ea0h :ear (or the ne=t si= :ears :ou 6ill re0ei?e 0ash (lo6s o( 7" . The initial in?estment is 74#4.3 . To the nearest per0ent, 6hat rate o( return are :ou e=pe0ting (rom this in?estment< ,&nnual Aompoun)ing. a. $; 8. "; 0. #2; ). 2#; e. # ;

Alass Botes

#$

*.

Jou are sa?ing up (or a )o6n pa:ment on a house. Jou 6ill )eposit 7! a month (or the ne=t 24 months in a mone: mar9et (un). Ho6 mu0h 6ill :ou ha?e (or :our )o6n pa:ment in 24 months i( the (un) earns # ; &3% 0ompoun)e) monthl:< a. 7#4,4$ 8. 7#5,$* 0. 7#2,"3 ). 7# ,5! e. 7 ",$"

$.

Jour mortgage pa:ment is 7! per month. There is e=a0tl: #$ pa:ments remaining on the mortgage. The interest rate s $. ;, 0ompoun)e) monthl:. The (irst pa:ment is )ue in e=a0tl: one month. 'hat is the 8alan0e o( the loan< 1Balan0e > 3+ o( remaining pa:ments.2 a. 7!2,*$4 8. 7**,2 5 0. 7!3,2 3 ). 7$2,5 2 e. 7$5,# * Jour mortgage pa:ment is 7*55 per month. /t is a 3 F:ear mortgage at ". ; 0ompoun)e) monthl:. Ho6 mu0h )i) :ou 8orro6< a. 7"3,$ 8. 7"*,2 0. 7"2,5 ). 7$5,# e. 7$",4

".

7ossibly Eew 7roblems.
# . 'hat is the ?alue o( the (ollo6ing set o( 0ash (lo6s to)a:< The interest rate is $.5;. Jear Aash Flo6 C F7#, a. 8. 0. ). e. 7$ 7 5*# 7# *2 7 "$* 7 52 #C 72 2C 74 3C 7! 4C 7$

##. The present ?alue interest (a0tor o( an annuit: )ue (or 3 :ears at $; eDualsC a. #4,#. $.3 8. #4,#.24. 0. 1# - #4,#. $. - #4,#. $.22 ). 1#4,#. $. - #4,#. $.2 - #4,#. $.32 e. Bone o( the a8o?e. #2. 'hat is the present ?alue o( 72,5 semiannual pa:ments re0ei?e) at the 8eginning o( ea0h perio) (or the ne=t # :ears< The &3% is !;. a. 3*,#"4.* 8. 3$,3 ".5 0. 35,$ ".5 ). 3!,$$4.$

Alass Botes

#"

#3. Jour mortgage pa:ment is 7! per month. There are e=a0tl: #$ pa:ments remaining on the mortgage. The interest rate s $. ;, 0ompoun)e) monthl:. The ne=t pa:ment is )ue imme)iatel:. 'hat is the 8alan0e o( the loan< 1HintC This is an annuit: )ue.2 a. 7!3,2 3 8. 7**,2 5 0. 7!2,*$4 ). 7$2,5 2 e. 7$5,# * #4. Jour mortgage pa:ment is 7! per month. There are e=a0tl: #$ pa:ments remaining on the mortgage. The interest rate s $. ;, 0ompoun)e) monthl:. The ne=t pa:ment is )ue in #5 )a:s. 'hat is the 8alan0e o( the loan< 1HintC &ssume 3 )a:s per month.2
a b c d e $62,993 $76,949 $62,576 $82,228 $84,825

#5. The present ?alue interest (a0tor o( an annual or)inar: annuit: (or 3 :ears at $; eDualsC a. #4,#. $.3 8. #4,#.24. 0. 1# - #4,#. $. - #4,#. $.22 ). 1#4,#. $. - #4,#. $.2 - #4,#. $.32 e. Bone o( the a8o?e. #!. The present ?alue interest (a0tor o( a semiannual or)inar: annuit: (or 3 :ears at $; eDualsC a 1#4,#. 4. - #4,#. 4.2 - #4,#. 4.32 8. 1#4,#. $. - #4,#. $.2 - #4,#. $.3 -#4,#. $.4 - #4,#. $.5 - #4,#. $.!2 0. 1#4,#. 4. - #4,#. 4.2 - #4,#. 4.3 - #4,#. 4.4 - #4,#. 4.5 - #4,#. 4.!2 ). 1#4,#. $. - #4,#. $.2 - #4,#. $.32 e. Bone o( the a8o?e. #*. The (uture ?alue interest (a0tor o( an or)inar: annuit: (or 3 :ears at $; eDualsC a. ,#. $.3 8. ,#.24. 0. 1# - ,#. $. - #. $.22 ). 1,#. $. - ,#. $.2 - ,#. $.32 e. Bone o( the a8o?e. #$. Suppose an annuit: 0osts 74 , an) pro)u0es 0ash (lo6s o( 7# , (ollo6ing eight :ears. 'hat is the rate o( return on the annuit:< a. ; 8. # .5; 0. #$.!; ). 25. ; e. 5 . ; #. & 2. & ##. A #2. B 3. A 4. E 5. D !. & *. B #3. & #4. & #5. D #!. A #*. D $. & #$. A o?er ea0h o( the

#ey:

". & # . B

Alass Botes Eotes on Time !alue of "oney -unctions in 8xcelQ.

2

Custom Time Value of o!e" fu!ctio!s a#e easil" do!e o! a s$#eads%eet. T%e follo&i!' fu!ctio!s ca! be i!se#ted i!to a s$#eads%eet. (!se#t me!u, )u!ctio!, )i!a!cial. ANNUITIES
• The (ollo6ing (un0tions appl: to annuitiesC 3+ F+ F+SAHEDGQE 33MT 3MT %&TE /3MT B3E%

SERIES OF UNEVEN PAYMENTS
B3+ /%% VB3+ V/%%

*ote+ a!" fi!a!cial fu!ctio!s #e,ui#e t%e -!al"sis Tool.a/ to be loaded. Tools menu. -dd i!s+ C%ec/ -!al"sis Tool.a/

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